Critical points of invariant functions on closed orientable surfaces

The relationship between critical points of equivariant functions and topo- logical invariants of an equivariant action on closed manifold is an interesting prob- lem.Inthispaper,westudythisrelationshipfororientation-preservingactionsoffinite groups G onaclosedorientablesurfaces.Wegiveanelementary,butdetailed,descrip- tion of the behaviour of the gradient field of an equivariant C 1 -function, we present an elementary, differential, proof of the Riemann-Hurwitz formula and we construct invariant C 1 -functions with the minimal number of critical orbits. These lead us to showthat,withafewexceptions,theequivariantLusternik-Schnirelmanncategoryofa closedorientabletopologicalsurfaceequalsthenumberofsingularorbitsoftheaction.